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RMP: Reduced-set matching pursuit approach for efficient compressed sensing signal reconstruction

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Journal of Advanced Research (2016) 7, 851–861

Cairo University

Journal of Advanced Research

ORIGINAL ARTICLE

RMP: Reduced-set matching pursuit approach for
efficient compressed sensing signal reconstructionq
Michael M. Abdel-Sayed, Ahmed Khattab *, Mohamed F. Abu-Elyazeed
Electronics and Communications Engineering Department, Faculty of Engineering, Cairo University, Giza 12613, Egypt

G R A P H I C A L A B S T R A C T

A R T I C L E

I N F O

Article history:
Received 19 April 2016
Received in revised form 6 August
2016
Accepted 26 August 2016
Available online 2 September 2016
Keywords:
Compressed sensing

A B S T R A C T
Compressed sensing enables the acquisition of sparse signals at a rate that is much lower than
the Nyquist rate. Compressed sensing initially adopted ‘1 minimization for signal reconstruction which is computationally expensive. Several greedy recovery algorithms have been


recently proposed for signal reconstruction at a lower computational complexity compared
to the optimal ‘1 minimization, while maintaining a good reconstruction accuracy. In this
paper, the Reduced-set Matching Pursuit (RMP) greedy recovery algorithm is proposed for
compressed sensing. Unlike existing approaches which either select too many or too few values per iteration, RMP aims at selecting the most sufficient number of correlation values per
iteration, which improves both the reconstruction time and error. Furthermore, RMP prunes

q A preliminary basic version of the RMP is accepted for presentation in IEEE International Conference on Image Processing (ICIP) 2016.
* Corresponding author. Fax: +202 3572 3486.
E-mail address: akhattab@ieee.org (A. Khattab).
Peer review under responsibility of Cairo University.

Production and hosting by Elsevier
http://dx.doi.org/10.1016/j.jare.2016.08.005
2090-1232 Ó 2016 Production and hosting by Elsevier B.V. on behalf of Cairo University.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).


852
Matching pursuit
Sparse signal reconstruction
Restricted isometry property

M.M. Abdel-Sayed et al.
the estimated signal, and hence, excludes the incorrectly selected values. The RMP algorithm
achieves a higher reconstruction accuracy at a significantly low computational complexity
compared to existing greedy recovery algorithms. It is even superior to ‘1 minimization in
terms of the normalized time-error product, a new metric introduced to measure the tradeoff between the reconstruction time and error. RMP superior performance is illustrated with
both noiseless and noisy samples.
Ó 2016 Production and hosting by Elsevier B.V. on behalf of Cairo University. This is an open
access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/

4.0/).

Introduction
In order to perfectly reconstruct a signal from its samples, the
signal is to be sampled at least at the Nyquist rate, which is
double the signal’s highest frequency component. However,
the Nyquist rate has two shortcomings. First, the Nyquist rate
of many contemporary applications is so high that it is too
expensive or even impossible to implement [1]. Second, the
large number of acquired samples are not fully used in the
reconstruction process or partially sacrificed. Recall that many
applications have to further compress the sampled signal for
efficient storage purposes or for transmission over a much limited bandwidth. For example, a typical digital camera has millions of imaging sensors, whereas the acquired image is usually
compressed into a few hundred kilobytes. Thus, a significant
amount of the acquired data – the least significant information
content – is sacrificed [2].
Recently, compressed sensing has presented itself as an efficient sampling technique that samples the signals at a much
lower rate compared to the Nyquist rate. Compressed sensing
simultaneously performs sensing and compression; thus, the
signal is sensed in a compressed form [1–7]. This results in a
considerable reduction in the number of measurements that
need to be stored and/or processed. Compressed sensing is
applicable to either sparse or compressible signals which typically have few significant coefficients in a suitable basis or
domain (e.g. Fourier and Wavelets). This includes a large variety of signals such as natural images, videos, MRI, and radar
signals [8]. The original signal can be recovered by convex optimization or greedy recovery algorithms.
Several greedy recovery algorithms have been recently
developed for sparse signal reconstruction [9–13]. These algorithms aim to reduce the computational complexity of the optimum ‘1 minimization, while maintaining a good
reconstruction accuracy. Such algorithms iteratively identify
the signal support (its nonzero indices) by correlating the measured signal with the sensing matrix columns. A number of
correlation values are selected in each iteration, and their

indices are added to a set of identified supports. Existing algorithms perform selection from the whole correlation vector,
which increases the reconstruction time. Furthermore, the
majority of the existing algorithms perform non-tunable selection, which results in selecting either too few or too many elements, causing larger reconstruction time and error.
In this paper, the Reduced-set Matching Pursuit (RMP), a
new thresholding-based greedy signal reconstruction algorithm
for compressed sensing is introduced by extending the algorithm in Abdel-Sayed et al. [14]. As a greedy recovery algorithm, RMP forms an estimate of the support of the sparse
signal in each iteration. Unlike the related algorithms, RMP

efficiently estimates the signal support by selecting values from
a reduced set of the correlation vector. Furthermore, the selection is performed in a signal-aware manner. That is, the number of selected elements per iteration changes based on the
distribution of the correlation values. Therefore, RMP targets
the selection of a sufficient number of elements per iteration.
The signal is then estimated using least square minimization
with nonzeros at indices from the identified support set. The
signal is then pruned to exclude the incorrectly selected elements. The residual is calculated from the pruned signal, and
the previous steps are repeated until a stopping condition is
met. Simulation results show that RMP has a high reconstruction accuracy at a significantly low computational complexity
compared to existing greedy recovery algorithms. Moreover,
RMP is capable of sparse signal reconstruction from noiseless
samples as well as from samples contaminated with additive
noise. More specifically, the normalized time-error product
of RMP is 87% to 95% less than that of ‘1 minimization at
high sparsity levels in the absence of noise. In the noisy samples case, the RMP normalized time-error product is 57% to
98% less than that of ‘1 minimization depending on the signal
to noise ratio (SNR).
Compressed sensing fundamentals
Consider a sparse signal x 2 Rn of sparsity level k. A measurement system that samples this signal to acquire m linear measurements is typically modeled as
y ¼ Ux;

ð1Þ


where U 2 R
is the sensing or measurement matrix, and
y 2 Rm is the measured vector or the samples.
Alternatively, the signal x may not be itself sparse, but it
may be sparse in a certain basis W, i.e. x ¼ Ws, where s is a
sparse vector. In this case, (1) is rewritten as
mÂn

y ¼ UWs ¼ As;

ð2Þ

where W is an n  n matrix which columns form a basis in
which x is sparse, and A ¼ UW is an m  n matrix.
Unlike legacy measurement systems, m is much less than n
in compressed sensing as the dimension of the measured vector
y is much lower than the dimension of the original signal x.
Yet, it was shown that the sparse (or compressible) signal x
can be recovered using the few measurements captured by y
provided that the sensing matrix satisfies the Restricted Isometry Property (RIP) [1,3].
A matrix A satisfies the restricted isometry property of
order k if there exists a dk 2 ð0; 1Þ such that
ð1 À dk Þkxk22 6 kAxk22 6 ð1 þ dk Þkxk22

ð3Þ


Reduced-set matching pursuit signal reconstruction


853

holds for all k-sparse signals x, where kxk2 is the ‘2 norm of the
signal x.
Random matrices of certain distributions satisfy the RIP
with high probability [15]. More specifically, if the entries of
a matrix are independent and identically distributed (i.i.d.)
and follow a Gaussian, Bernoulli or sub-Gaussian distribution,
the probability that the matrix does not satisfy the RIP is
exponentially small.
The natural, and the most straightforward, approach to
recover a sparse signal from a few set of measurements is by
solving an ‘0 norm optimization problem. However, the objective function of the ‘0 optimization problem is nonconvex, and
hence, finding the solution that approximates the true minimum is NP-hard [4]. One way to transform this NP-hard problem into something more tractable is to replace the ‘0 norm
with its convex approximation ‘1 norm. In this case, the transformed problem can be solved as a linear program.
Donoho [4] suggested minimizing the ‘1 norm k Á k1 to
reconstruct the sparse signal as follows:
x^ ¼ arg minkzk1 subject to y ¼ Uz:

ð4Þ

z

In practice, the measured samples are typically contaminated with additive noise. In this case, the measured vector
is given by
y ¼ Ux þ e;

ð5Þ

where e is the sample noise and kek2 < . ‘1 minimization can

still be used to reconstruct the original sparse signal x with an
error that cannot exceed the noise level  as follows [16]:
x^ ¼ arg minkzk1 subject to ky À Uzk2 6 :

ð6Þ

z

In both the noiseless sample and noisy sample cases, ‘1 minimization is a powerful solution for the sparse problem. However, this solution is computationally expensive [1].

Fig. 2

Classification of sparse recovery algorithms.

2. Selection and support merging: One or more of the elements
of g with the largest absolute values are selected in each
iteration. The indices of the selected elements are merged
into the identified support set which is used to approximate
the signal.
3. Signal estimation: The sparse signal is estimated based on
the identified support using least square minimization.
Some algorithms (thresholding-based algorithms) perform
a pruning step to the estimated signal, keeping only the k
largest absolute values of the signal, and setting the rest
to zeros.
4. Residual calculation: The residual is calculated based on the
estimated signal.
Greedy recovery algorithms can be classified into thresholdless algorithms and thresholding-based algorithms depending
on whether or not they prune the estimated signal by applying
a hard thresholding operator. In what follows, the main existing algorithms in each category are discussed and summarized

in Fig. 2.

Greedy recovery algorithms
Threshold-less greedy recovery algorithms
Motivated by the need to develop computationally inexpensive
solutions, various greedy algorithms have been proposed in the
literature for signal recovery. Greedy recovery algorithms iteratively attempt to find the signal support. In each iteration, the
sparse signal is estimated based on the identified support set
through least square minimization. Fig. 1 shows a generic
block diagram of the main steps for such greedy algorithms.
The function of each block is briefly described as follows:
1. Correlation: The residual r is correlated with the columns of
the sensing matrix U to form a proxy signal g.

Fig. 1

General block diagram of recovery algorithms.

The first greedy recovery algorithm is the Basic Matching Pursuit (BMP) [1,17]. BMP selects only one element from the correlation vector per iteration, and adds its index to the identified
support set. However, the residual is calculated without performing least square minimization, which results in higher
reconstruction error. Another simple greedy recovery algorithm is the Orthogonal Matching Pursuit (OMP) [9,18].
OMP performs least square minimization to estimate the signal, which results in improvement over BMP. However,
OMP selects only one element from the correlation vector
per iteration as in BMP. For a k-sparse signal, OMP needs k
iterations in order to reconstruct the signal.
Alternatively, other algorithms add more than one index
per iteration, resulting in a faster convergence time. For
instance, the Generalized Orthogonal Matching Pursuit
(GOMP) selects a fixed number of elements per iteration
[10]. Meanwhile, the Regularized Orthogonal Matching Pursuit (ROMP) chooses a set of k largest nonzero elements, then

divides them into groups of comparable magnitudes and
selects the group of maximum energy [19,20]. The Stagewise
Weak Orthogonal Matching Pursuit (SWOMP) selects the elements with absolute values larger than or equal to amaxl jgl j,
where 0 < a < 1 and maxl jgl j is the largest magnitude element


854
in the correlation vector [21]. The Stagewise Orthogonal
Matching Pursuit (StOMP) [22] selects the elements larger than
a certain configurable value determined by the constant false
alarm rate (CFAR) strategy originally developed for radar systems [11].
Other algorithms exploit the structure of the signal sparsity
such as the Tree-based Orthogonal Matching Pursuit (TOMP)
[23–25]. On the other hand, the Multipath Matching Pursuit
models the problem of finding the candidate support of the signal as a tree search problem [26]. Finally, it is worth mentioning that some algorithms that fall under this category speed up
the minimization step using iterative matrix inversion techniques [27].
Drawbacks of threshold-less greedy algorithms
Since BMP and OMP add only one index per iteration, they
require a larger number of iterations than the rest of the algorithms. While ROMP improves the speed of OMP by selecting
multiple elements per iteration, its reconstruction error is larger, especially for higher sparsity levels. The algorithm often
results in adding a larger number of indices per iteration than
is necessary, which usually includes ones not belonging to the
support of the original signal. SWOMP and StOMP attempt to
improve the selection stage by using different selection strategies. However, SWOMP still suffers from the same drawback
of ROMP. Meanwhile, StOMP gives closer error performance
to OMP, while requiring less execution time for higher sparsity
levels. It is worth noting that none of the aforementioned algorithms contain a pruning step. Thus, incorrectly selected
indices will appear in the signal estimate, which degrades the
performance reflected by a deterioration in the reconstruction
accuracy.

Thresholding-based greedy recovery algorithms
A common drawback in all the aforementioned greedy algorithms is that if an incorrect index is added to the support
set in a certain iteration, it remains in all subsequent iterations,
possibly degrading the performance. Thresholding-based algorithms handle this problem by applying a hard thresholding
operator which removes one or more of the indices having
the least energy from the identified support set. An example
is the Compressive Sampling Matching Pursuit (CoSaMP)
[12], which selects 2k elements per iteration and performs
pruning after signal estimation. The Subspace Pursuit (SP) is
another thresholding-based algorithm which selects k elements
per iteration [13]. Pruning is then performed, followed by an
extra least square minimization step. Iterative Hard Thresholding (IHT) is another thresholding-based recovery algorithm
which recursively solves the sparse problem while applying the
hard thresholding operator [28,29].
Drawbacks of thresholding-based greedy algorithms
Thresholding-based algorithms such as CoSaMP and SP add a
pruning step at the end of each iteration. However, such algorithms select a fixed number of elements per iteration (e.g. 2k
in CoSaMP and k in SP). Such a selection is constant for all
iterations and does not adapt to the distribution of the values
of correlation. Furthermore, it usually results in selecting too

M.M. Abdel-Sayed et al.
many elements causing a larger reconstruction time, since
more than necessary components are sorted in each iteration.
A large and fixed selection further increases the iteration time
as more than necessary nonzero values have to be estimated by
least square minimization. Selecting too many elements also
reduces the accuracy of the signal estimate, especially for larger
sparsity and when working on a noisy measurement, when
incorrect indices are selected and kept through the subsequent

pruning steps. Finally, the iterative nature combined with sacrificing the least square minimization step in the IHT algorithm results in an increased reconstruction time and error.
The rest of this paper is organized as follows. The RMP
algorithm is proposed in the ‘‘Reduced-set Matching Pursuit”
Section, and thoroughly evaluates its different performance
aspects in the ‘‘Performance Evaluation and Discussions” Section. Section ‘‘Conclusions” concludes the paper.
Reduced-set matching pursuit
In this section, the Reduced-set Matching Pursuit (RMP), a
thresholding-based greedy recovery algorithm is presented.
RMP main goal is to reconstruct a sparse signal x from measurements given by (1) or (2) as accurately and efficiently as
possible. In order to achieve these goals RMP performs 4 main
steps. First, RMP iteratively identifies the support of the sparse
signal by appropriately selecting elements from a significantly
reduced set of the correlation values. This contrasts with existing algorithms in which the selection is performed from the
whole correlation vector and is performed in a signalagnostic manner in the majority of existing algorithms. Second, RMP estimates the sparse signal based on the identified
support set. Even though RMP uses least square minimization
to estimate the signal, its convergence time is much less than
existing techniques since RPM least square minimization targets a significantly reduced set of indices. Third, RMP uses
pruning to exclude the incorrectly selected elements, and
hence, prevent such erroneous selections from degrading the
performance. Fourth, a residual is then calculated to remove
the estimated part from the measurement vector. These steps
are repeated until a stopping criterion is met.
RMP components
In what follows, the four main components of the RMP algorithm are explained in detail.
Support identification
In order to reconstruct the sparse signal, its support (nonzero
indices) needs to be identified. This is done iteratively, where in
each iteration the identified support set is updated. First, the
measured vector y is correlated with the columns of the sensing
matrix U to obtain a correlation vector g. The non-zero indices

of the sparse signal are expected to have relatively large magnitudes of correlation. Thus, some of the highest magnitude
elements of the correlation vector are selected according to a
specific ‘‘selection strategy”. The indices of the selected elements are merged with the identified support set.
The selection strategy is one of the main factors on which
the performance of the recovery algorithm depends. The selection stage should be able to select elements corresponding to


Reduced-set matching pursuit signal reconstruction

855

nonzero indices of the original sparse signal. It should not
select too few elements, which leads to an excessively large
number of iterations, which in turn causes a larger reconstruction time. Nor should it select too many elements, which leads
to performing calculations on a much larger amount of data
(which includes sorting, matrix inversion, and least square
minimization). Not only does this increase the reconstruction
time, but it also causes the selection of elements which indices
do not belong to the support of the original signal, which leads
to an increase in the reconstruction error. Therefore, it is necessary for the algorithm to achieve a compromise in the number of selected elements per iteration. Existing techniques
either select too few elements [9,10,18] or too many elements
[12,13,19,20,22], which increases their reconstruction time or
reduces their reconstruction accuracy respectively.
In contrast, RMP targets the selection of a sufficient
enough number of elements using a double thresholding technique. RMP selects the indices which most likely belong to the
support of the original signal, without taking too few or too
many indices per iteration. Based on the distribution of the
absolute values of g, the number of selected elements is not
constant for all iterations (even though a and b are constants).
For steeper distributions of the absolute values of g, fewer elements are selected. For flatter distributions, more elements are

selected.
RMP achieves this goal in two steps. First, the elements
from which selection is performed are reduced to a set containing the bk top magnitude elements. Then, elements whose
magnitudes are larger than a fixed fraction 0 < a < 1 of the
maximum element are selected from the reduced set, and their
indices are added to the support set. The proper selection of
the constant values of the a and b parameters leads to the
selection of an optimum number of elements per iteration,
which in turn contributes to a high reconstruction accuracy
and a low reconstruction complexity.
Signal estimation
After the selection and support merging stage, a new signal
estimate x^ is formed based on the merged support set. This
is performed using least square minimization. That is, the algorithm finds the signal x^ which minimizes ky À U^
xk2 while having non-zeros at the indices obtained from the identified
support set. Such minimization is done via the multiplication
of the pseudo-inverse given by
À1

UyT ¼ ðUTT UT Þ UTT ;

ð7Þ

where UT is a matrix that contains the columns of U with
indices in the identified support set T. It should be noted here
that the calculation of the pseudo-inverse requires the inversion of a matrix whose size is dependent on the number of
indices in the identified support set. Since RMP selects an optimum number of elements per iteration, which is much smaller
than that selected by other existing algorithms, the size of the
matrix is smaller, and the reconstruction is faster.
Pruning

Next, the estimated signal is pruned. Pruning is a technique
that is used to enhance the performance of recovery algorithms
[12]. Recovery algorithms inevitably select one or more

elements whose indices do not belong to the support set of
the original signal during the reconstruction process. Without
pruning, such elements remain in the signal estimate during the
consecutive iterations, which reduces the reconstruction accuracy. Hence, convergence is slower and the reconstruction time
is generally affected.
In RMP, the estimated signal is pruned by removing the
elements which have the least contribution to the estimated signal from the identified support set. RMP only keeps those corresponding to the k largest magnitude components of the
estimated signal. The benefit of the pruning step is even more
evident in the reconstruction of signals from samples contaminated with noise.
Residual calculation
A residual is then calculated by subtracting the contribution of
the estimated signal from the measured vector. The residual is
given by
r ¼ y À U^
x:

ð8Þ

This residual is then correlated with the columns of the
sensing matrix for the successive iterations. The previous steps
are repeated until a stopping criterion is met. RMP terminates
if the norm of the residual is less than 1 or if the difference
between the norms of the residuals in two successive iterations
is less than 2 , whichever occurs first. Otherwise, a maximum of
k iterations are performed.
RMP algorithm

Initially, the signal estimate is set to a zero vector and the
residual to the measured vector y. In each iteration, the following steps are performed:
1. Signal proxy formation: A signal proxy, g, is formed by correlating the residual with the sensing matrix columns.
2. Selection and support merging: The vector g is sorted in a
descending order of absolute values. The elements whose
absolute values are larger than or equal to a maxl jgl j, where
0 < a < 1, are selected from a reduced set containing the bk
largest magnitude elements. The indices of the selected elements are united with the already identified support set.
3. Signal estimation: An estimate of the signal is formed by
least square minimization. This is done via multiplication
by the pseudo-inverse of the sensing matrix.
4. Pruning: The k largest magnitude components in the signal
estimate are retained. The rest are set to zero.
5. Residual calculation: The new residual is calculated from
the pruned signal.
At the end of each iteration, the RMP algorithm checks
whether the norm of the residual is less than 1 or whether
the difference between the norms of the residuals in two successive iterations is less than 2 . If either condition is met, the
RMP algorithm terminates. Otherwise, RMP terminates after
a maximum of k iterations.
Algorithm 1 summarizes the RMP algorithm. The operator
Lk ðÁÞ returns the index set of the k largest absolute values of
the elements of its argument vector. The hard thresholding


856

M.M. Abdel-Sayed et al.

operator Hk ðÁÞ retains only the k elements with the largest

absolute values and sets the rest to zero.
Algorithm 1. Reduced-set Matching Pursuit.
Input: Sensing matrix U, measurement vector y, sparsity level k,
parameters a and b.
Initialize: x^½0Š ¼ 0; r½0Š ¼ y; T½0Š ¼ £.
for i ¼ 1; i :¼ i þ 1 until the stopping criterion is met do
g½iŠ
Uà r½iÀ1Š {Form signal proxy}
J
Lbk ðg½iŠ Þ {Indices of bk largest magnitude elements in g}
W

½iŠ

½iŠ

fj : jgj j P a maxjgl j; j 2 Jg {Indices of elements in J larger
l

½iŠ

than or equal to a maxjgl j}
l

T

W [ suppð^
x½iÀ1Š Þ {Support merging}

bjT


UyT y; bjTc

^½iŠ

Hk ðbÞ {Prune approximation}

x

0 {Signal estimation}

r
y À U^
x½iŠ {Update residual}
end for
Output: Reconstructed signal x^

The effect of a and b
The performance of the RMP algorithm is governed by the
proper selection of its a and b parameters. Here, the effect of
a and b on the performance of RMP is discussed. In the Performance Evaluation Section, simulations are used to obtain
their best value ranges and verify that the RMP algorithm performance is not sensitive to a particular choice in such a range.
There are three different ranges for a for which the performance drastically changes.
First, when a is very small and close to zero, all the elements
in the reduced set are selected. Having large values of b in this
case may improve the performance, but will cause a larger
reconstruction time. This is due to the selection of a larger
number of indices per iteration than what is necessary. For
small a and for small values of b, the reconstruction error is
larger, since a very small number of indices are selected, which

is not enough to select the correct support of the signal. Furthermore, a larger number of iterations are required, which
in turn leads to a larger reconstruction time.
Second, for larger values of a close to 1, the number of
selected indices per iteration is too small. Thus, a large number
of iterations are required and the reconstruction time is larger
regardless the value of b.
Third, when a is neither too close to 0 nor too close to 1, the
best compromise is achieved. The number of selected elements
per iteration are neither too large (as in the first case) nor too
small (as in the second one). Such a moderate choice of a will
also relax the requirements on b which will also tend to be
moderate as there will be no need to select a large number of
indices. This leads to improvements in the reconstruction time
and accuracy. Simulation results show that the exact choice of
the a and b values in this moderate range does not significantly
affect the performance.
Noise robustness
In many signal reconstruction applications, the measured samples are contaminated with additive white noise. Therefore, it

is necessary for the recovery algorithm to be able to reconstruct the sparse signal from noisy samples as accurately as
possible. Next, the reconstruction capability of RMP when
the measured samples are contaminated with additive white
noise as given by (5) is discussed.
Since the measured signal y is contaminated with noise, the
correlation vector g is noisy as well. This may result in the
selection of incorrect elements from g in some iterations,
depending on the signal-to-noise ratio (SNR). The higher the
SNR, the higher the probability of selecting incorrect elements,
and vice versa. Consequently, a signal estimate is formed with
some elements of the support set at incorrect indices. Now, if

the recovery algorithm does not have a pruning step, there is
no way to exclude such elements from the identified support
set, and the performance of the algorithm will deteriorate.
On the other hand, algorithms which have a pruning step, such
as RMP, are capable of excluding incorrectly added elements
in each iteration, and iterating until the correct ones are found.
Thus a more accurate estimate of the support set is generated,
and consequently a more accurate estimate of the signal is
formed. Such incorrectly identified elements are pruned with
high probability after the signal estimate is formed, since they
have the least contribution to the original signal.
Furthermore, RMP selects a smaller number of elements
per iteration, compared to other thresholding-based algorithms that perform pruning, causing its performance to be
more robust in the presence of noise. This is because selecting
a larger number of noisy elements than is necessary per iteration (as the case with other related algorithm) makes such
algorithms more error-prone. Recall that the pruning step
excludes the elements of the support set which have the least
contribution to the estimated signal. When there are too many
elements present in the noisy signal estimate, pruning may
keep some of the incorrectly added ones due to noise. This
results in a larger error for lower SNR levels for such algorithms. Therefore, RMP outperforms other thresholdingbased algorithms in applications that suffer from noise.

Performance metrics
In the next section, the performance of RMP against existing
related techniques as well as the original ‘1 minimization is
evaluated. The used performance metrics are as follows:
 The reconstruction time t in seconds, which is the time
required to reconstruct the sparse signal from the measurement signal.
 The reconstruction error e, which is the reconstruction error
relative to the ‘2 norm of the signal defined as

kx À ^xk2 =kxk2 .
 We introduce the normalized time-error product in which the
product of the time and error of each algorithm is normalized
over the largest product value of all algorithms, that is:
Normalized time À error product ¼

tij Á eij
;
maxi;j ftij Á eij g

ð9Þ

where, tij and eij are the reconstruction time and reconstruction error of algorithm i at sparsity level j, respectively. This
metric accounts for the trade-off between time and error,
since some algorithms give higher reconstruction accuracy
at the expense of higher computational complexity.


Reduced-set matching pursuit signal reconstruction

857
Reconstruction error

Reconstruction time (sec)
0.2

0.15

0.15
0.1

0.1
0.05

0.05

0
2

0
2

1

1
0

0

0.4

0.2

0.8

0.6

1
0

Number of iterations


0.2

0

0.4

0.6

0.8

1

Selected elements per iteration

80

150

60

100

40
50
20
0
2

0

2

1

1
0

0

0.2

0.8

0.6

0.4

1
0

0.2

0

0.4

0.6

0.8


1

Normalized time-error product
1

0.5

0
2

1

0

0

0.2

0.4

0.6

0.8

1

Fig. 3 Impact of a and b on (a) reconstruction time, (b) reconstruction error, (c) number of iterations, (d) the average number of selected
elements per iteration, and (e) Normalized time-error product at a sparsity level of 70.

Other metrics are also considered that help understand the

differences in the dynamics of how each algorithm reconstructs
the original signal such as:
 The number of iterations performed by the algorithm.
 The average number of selected elements per iteration.
 The average size of the merged support set. For
thresholding-based algorithms, this is taken before pruning
for the sake of fairness in comparison.

Performance evaluation and discussions
Simulation setup
In this section, the performance of the proposed RMP algorithm against the performance of the following algorithms: ‘1
minimization, OMP, ROMP, IHT, SWOMP, StOMP, SP,
and CoSaMP is illustrated via MATLAB simulations. For
each algorithm, the reported results are the average of the


858

M.M. Abdel-Sayed et al.
0.5
OMP
SWOMP
SP
CoSaMP
RMP

Time (sec)

0.4


0.3

0.2

0.1

0
20

40

60

80

100

120

140

Sparsity

(a) Reconstruction time
2

The effect of a and b
L1 Norm
OMP
SP

CoSaMP
RMP

Error

1.5

1

0.5

0
20

40

60

80

100

120

140

Sparsity

(b) Reconstruction error
Normalized Time-Error Product


0.03
L1 Norm
SWOMP
SP
CoSaMP
RMP

0.025
0.02
0.015
0.01
0.005
0
20

40

60

80

100

120

140

Sparsity


(c) Normalized time-error product
Fig. 4

The sensing matrix U of dimensions m  n is randomly generated from i.i.d. Gaussian distribution with columns having
unit ‘2 norm.
For SWOMP, a ¼ 0:7 is used, which is the same value used
in [21]. For IHT, the step size, l, is tuned by obtaining the metrics at a sparsity level of 70 using values of l ranging from 0.1
to 1 with 0.1 steps. It was found that l ¼ 0:3 results in the least
normalized time-error product; therefore, this value is used for
IHT in the following simulations. For StOMP, the implementation that is available as a part of the SparseLab Toolbox for
Matlab is used.
For the noiseless case, the results of the different metrics for
sparsity levels ranging from 10 to 150 are reported. For the
noisy case, AWGN is added to the measured samples at different values of SNR. The results of the metrics against SNR
from À10 dB to 50 dB at a sparsity level of 70 are reported.

Performance attributes for the noiseless case.

metrics evaluated for 100 independent trials. In each trial, a
random sparse signal of length n ¼ 1000 of uniformly distributed integers from 0 to 100 is generated. This paper only
presents the results of m ¼ 250 measurements. The results of
other values of m are omitted since similar observations were
obtained. The only difference is that as m increases (or
decreases), the errors occur at higher (or lower) sparsity levels.

Before comparing the performance of RMP against the other
existing algorithms, the effect of its a and b parameters is studied first to obtain their best values. In order to study the effect
of the a and b parameters, the value of a is varied from 0.1 to 1
with 0.1 steps, and the value of b from 0.05 to 2 with 0.1 steps.
The different performance aspects (namely, reconstruction

time, error, the number of iterations, the number of selected
elements per iteration, and the normalized time-error product)
metrics are depicted for the different pair of (a; b) in Fig. 3(a)
to (e), respectively. These results are averaged over 100 independent trials per (a, b) pair at different sparsity levels. Only
the results at a sparsity level of 70 are reported here. However,
similar results and conclusions were obtained at the other sparsity levels.
For smaller values of a up to 0.5, values of b larger than
0.75 cause larger reconstruction time, as shown in Fig. 3(a).
As explained in the previous section, a larger number of indices
per iteration are selected as illustrated in Fig. 3(d). For very
small values of b with small a value, the reconstruction error
is larger as depicted in Fig. 3(b). A very small number of
indices are selected and a larger number of iterations are
required, as shown in Fig. 3(c), which in turn leads to a larger
reconstruction time. For such low values of a, values of b ranging from about 0.15 to 0.75 give the smallest normalized timeerror product as depicted in Fig. 3(e).
In the other end of values of a ranging from 0.8 to 1, the
number of selected indices per iteration is too small. Thus, a
large number of iterations are required, and hence, the reconstruction time is larger.
In contrast, values of a ranging from 0.5 to 0.7 give the best
performance compromise. The number of selected elements
per iteration is neither too large, as in the first range, nor
too small, as in the second one. For this range, b ranging from
about 0.15 to 0.75 gives the smallest normalized time-error
product.
It is noted that the performance of the algorithm is not very
sensitive to the values of a and b as long as they are in the
aforementioned optimum range. It can be also noted that as
the value of a increases, the effect of b becomes less evident.
This is due to the fact that the number of selected indices is
mainly limited by a in this case. Similar results are obtained

for sparsity levels ranging from 50 to 100. The values a ¼ 0:7


Reduced-set matching pursuit signal reconstruction
Table 1

859

Normalized time-error product Â100 (noiseless case).

Sparsity

60

70

80

90

100

110

120

130

140


150

L1 Norm

0.00

0.00

0.00

0.10

1.09

2.24

3.25

4.02

4.32

4.95

OMP

0.01

0.05


0.20

0.48

0.90

1.31

1.62

1.92

2.37

2.86

ROMP

0.05

0.20

0.33

0.25

0.27

0.31


0.27

0.27

0.25

0.27

IHT

0.25

0.55

0.89

1.16

1.43

1.80

2.04

2.39

2.77

3.17


SWOMP

0.00

0.01

0.13

0.26

0.35

0.39

0.37

0.40

0.43

0.45

StOMP

0.00

0.02

0.11


0.21

0.26

0.30

0.31

0.31

0.29

0.30

SP

0.00

0.00

0.04

0.20

0.64

2.15

8.09


12.75

16.53

21.80

CoSaMP

0.00

0.01

1.62

100

21.96

23.28

27.23

29.93

34.27

39.53

RMP


0.00

0.00

0.03

0.09

0.14

0.18

0.21

0.22

0.24

0.27

The highlighted cells represent the least normalized time-error product.

and b ¼ 0:25 are selected to be used in the rest of the
simulations.
Performance comparison
In what follows, the simulations results that demonstrate the
performance advantages of RMP compared to other existing
algorithms are presented. While the presented plots only show
the results of the most relevant algorithms, the results of all the
algorithms are also tabulated for interested readers.

Noiseless case
First, the case in which the signal is not contaminated with
noise is considered. Fig. 4(a) depicts the reconstruction time
versus the signal sparsity level. ‘1 minimization is omitted since
it takes considerably longer time. The proposed RMP has the
least reconstruction times. This is due to the selection of a just
sufficient number of elements per iteration. SWOMP and
ROMP achieve slightly higher reconstruction times. It should
be noted that both SWOMP and ROMP are not
thresholding-based (i.e., they do not perform pruning) which
causes larger reconstruction error. The reconstruction time of
other thresholding-based algorithms increases rapidly at sparsity levels of 70 for CoSaMP and 100 for SP. This is due to the
selection of a larger number of elements.
Fig. 4(b) shows the reconstruction error as a function of the
sparsity level. For low sparsity levels, most of the algorithms
produce very low errors, giving accurate signal estimates.
However, as the sparsity of the signal increases, the differences
between the reconstruction capability of the algorithms start to
become significant. The optimal ‘1 minimization has the least
error – despite its extremely long reconstruction time. The proposed algorithm, RMP, has the lowest error compared to all
other greedy algorithms for most of the sparsity levels. However, beyond a sparsity level of about 100, the error for all
algorithms is too large to be used in practical applications.
The proposed normalized time-error product metric captures both performance aspects. Fig. 4(c) shows the normalized
time-error product as a function of sparsity. RMP has the

smallest product for most sparsity levels except for sparsity
levels around 80 where ‘1 minimization is slightly smaller. This
means that RMP achieves a high reconstruction accuracy at
low complexity compared to other algorithms including ‘1 minimization (which achieves slightly higher accuracy but at the
expense of significantly longer time). Table 1 lists the normalized time-error product of all the simulated algorithms for

noiseless samples.
Noisy case
Next, the case in which the signal is contaminated with additive noise is considered. Fig. 5(a) depicts the reconstruction
time versus the SNR for the noisy case. RMP has the least
reconstruction time for all values of SNR values. Again the
graph for ‘1 minimization is omitted since it is considerably
higher than the rest of the algorithms.
Fig. 5(b) illustrates the error for the noisy case. ‘1 minimization has the lowest error for higher values of SNR, followed by
RMP. For lower SNR, RMP and SP give the least error. It can
be seen that SWOMP, StOMP, and ROMP have high reconstruction error, especially at lower values of SNR. This is
due to the fact that they do not perform pruning. While
CoSaMP performs pruning, the large number of selected elements per iteration makes it more error-prone.
Fig. 5(c) shows the normalized time-error product for the
noisy case. As with the noiseless case, RMP has the smallest
product for all SNR levels in the noisy case. This implies that
RMP is more robust against noise compared to the rest of the
algorithms as it has a high reconstruction accuracy at a low
complexity – even under low SNR levels. Table 2 lists the full
normalized time-error product of all the simulated algorithms
for noisy samples.
Dynamics of different algorithms
Finally, the dynamics of the different algorithms are discussed
in order to better explain how RMP achieves its outstanding
performance. More specifically, the number of iterations taken
by each algorithm for the noiseless case, the average number of


860

M.M. Abdel-Sayed et al.

0.6
0.5

Time (sec)

Table 2

OMP
SWOMP
SP
CoSaMP
RMP

0.4

Normalized time-error product Â100 (noisy case).

0.3
0.2
0.1
0
-10

0

10

20

30


40

50

SNR

The highlighted cells represent the least normalized time-error
product.

(a) Reconstruction time
0.35
L1 Norm
OMP
SP
CoSaMP
RMP

0.3

Error

0.25
0.2
0.15
0.1
0.05
0
-10


0

10

20

30

40

50

SNR

(b) Reconstruction error

OMP. However, the fact that none of the aforementioned
threshold-less algorithms perform pruning leads to a larger
error.
Next, the SP, CoSaMP, and RMP thresholding-based algorithms are studied. CoSaMP has the largest merged support set
size, followed by SP. This not only causes a larger reconstruction time, but also causes a larger reconstruction error, especially for higher sparsity levels. On the other hand, the
selection strategy of RMP results in adding a much smaller
number of indices per iteration. This keeps the support set size
significantly smaller in successive iterations, giving a relatively
lower time and error. While RMP requires a larger number of
iterations up to about a sparsity level of 70, the operations are
performed on a much smaller amount of data. The overall
result is a high reconstruction accuracy at a lower complexity.
Conclusions


Normalized Time-Error Product

0.05

0.04

L1 Norm
SWOMP
SP
CoSaMP
RMP

0.03

0.02

0.01

0
-10

0

10

20

30

40


50

This paper has introduced RMP: a new thresholding-based
greedy algorithm for signal recovery for compressed sensing
applications. RMP targets the selection of just a sufficient
number of elements per iteration. This is performed by appropriately selecting elements from a reduced set of correlation
values. Pruning is then performed to exclude incorrectly
selected elements. Simulation results for both the noiseless
and noisy cases have shown that the proposed RMP algorithm
is superior to the main existing greedy recovery algorithms
both in terms of reconstruction time and accuracy. Furthermore, RMP is even superior to ‘1 minimization in terms of
normalized time-error product, a measure which accounts for
the trade-off between the reconstruction time and error.

SNR

(c) Normalized time-error product
Fig. 5

Performance attributes for the noisy case.

selected elements per iteration, and the average size of the
merged support set before pruning are investigated.
OMP selects one element per iteration and performs a number of iterations equal to the sparsity level, thus taking a relatively large reconstruction time. Meanwhile, ROMP and
SWOMP select a larger number of elements without pruning,
thus performing a much smaller number of iterations and
requiring much lower reconstruction time. By design, StOMP
performs a maximum of a fixed number of iterations, which
is set to 10. This leads to a lower reconstruction time than


Conflict of interest
The authors have declared no conflict of interest.
Compliance with ethics requirements
This article does not contain any studies with human or animal
subjects.
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